Property $(\beta )$ and uniform quotient maps

In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether a Banach space that is a uniform quotient of ${\mathrm{\ell }}_p$, $1<p\mathrm{\ne }2<\mathrm{\infty }$, must be isomorphic to a linear quotient of ${\mathrm{\ell }}_p$. We apply the geometric property $(\beta )$ of Rolewicz to the study of uniform and Lipschitz quotient maps, and answer the above question positively for the case $1<p<2$. We also give a necessary condition for a Banach space to have $c_0$ as a uniform quotient.